1. Field of the Invention
This invention relates to a method and an apparatus for generating an image representing an iso-valued surface, i.e. a set of points with a specified scalar value, from a set of points each having a scalar value, where the points are distributed in three dimensional space. The points and associated scalar values are generated by methods such as the three-dimensional finite element method (3D FEM).
2. Prior Art
"Volume Rendering" has recently been studied in the field of medical image processing, and has some application to the visualization of numerical simulation results. Though volume rendering is applicable to the three-dimensional finite difference method (3D FDM), it cannot be applied directly to the 3D FEM, which has no regular connections among neighboring grids (elements), because volume rendering assumes that a scalar field is defined on orthogonal grids.
The FEM divides an area to be analyzed into polyhedrons called elements, approximates the potential distribution in each element to a simple function, and obtains a scalar value at each node point. Generally, this method defines node points P irregularly, depending upon the nature of the area to be analyzed, as shown in FIG. 10.
When the volume rendering method is applied to the FEM output, scalar values defined on irregular grid points (node points) must be mapped to a regular grid Q (hexahedron). Information contained in the original 3D FEM analysis results may be lost, as in the case of the node points in region R of FIG. 10, unless the grid to which the 3D FEM results are mapped is positioned appropriately. Not a little information is lost in mapping, because the appropriateness depends heavily on the arrangement of the irregular grid of the FEM.
One method of extracting iso-valued surfaces without mapping node point information to a hexahedral grid is presented in the inventor's paper, "Visualization of Equi-Valued Surfaces and Stream Lines," I-DEAS/CAEDS International Conference Proceedings, Oct. 1988, pp. 87-97. This method is also disclosed in "Method for Reconstructing Solid Elements into Linear Tetrahedral Elements," IBM Technical Disclosure Bulletin, 06-89, pp. 340-342. The method in these papers creates a node point and a scalar value for it at each of the face centers and the volume center of each volume element, and then divides each volume element into linear tetrahedral elements defined by the node points. FIG. 11 shows the division of a parabolic wedge element into thirty-six linear tetrahedral elements.
FIG. 12 shows one example of a created tetrahedral element. Assume that the scalar values of vertices P, Q, R, and S are 8, 2, 4, and 0, respectively. Visualization of the iso-valued surface with the scalar value 6 is considered below. Points X, Y, and Z, where the scalar value is 6, are searched by linear interpolation. The triangle defined by those three points X, Y, and Z is the approximation of the iso-valued surface within the tetrahedron PQRS.
The prior art method obtains geometric data of the iso-valued surface approximated by triangles, in other words, position data and connection data of the triangles, and performs shading on the basis of the data. However, the volume of the geometric data of the triangles is apt to be enormous. In particular, when multiple iso-valued surfaces are visualized as semi-transparently shaded images, a large volume of computer storage is required, and considerable time is required to generate the image data, because the geometric data of the triangles must be generated for each iso-valued surface.